ABSTRACT
Magnetization or spin dynamics is usually described phenomenologically by means of the well-known Landau-Lifshitz-Gilbert (LLG) equation [1] - [4]:
δ δt M(r,t) = −γM(r,t)×Heff (r)+ λ
m M(r,t)×{M(r,t)×Heff (r)} , (9.1)
where γ is the gyromagnetic ratio, λ the Gilbert damping factor, and the effective forcing field is given by the derivative of the internal energy E(M(r,t)) with respect toM(r,t),
Heff (r) = δE(M(r,t))/δM(r,t) . (9.2)
For simulation purposes very often Heff (r) is decomposed into various terms,
Heff (r) = Hextern(r)+Hexch(r)+Hanis(r)+Hdipole(r)+Helast(r) , (9.3)
corresponding in turn to contributions from applied external fields, intrinsic exchange interactions, the magneto crystalline anisotropy, magnetic dipoledipole interactions, and magneto-elastic effects. For practical purposes the individual terms in Eq. (9.3) are then taken
either from experiment or by adopting parameters listed in the literature.
